Mary Shelley, as noted in the preface of the Dover Thrift Edition of Frankenstein, was the daughter of renowned writers William Goodwin and Mary Wollstonecraft. William Goodwin was known for his outspoken views on utilitarianism and anarchism. Mary Wollstonecraft, on the other hand, was also a writer (much of her writing focusing upon women's rights). Therefore, Mary Shelley's influence to write came at a young age.
Outside of the influence of her parents, Mary Shelley was also influenced by her husband, who she married in 1816. Shelley's husband was the Romantic poet Percy Bysshe Shelley.
The story of the creature's inception is almost as famous among Romantic and Gothic enthusiasts. Mary and Percy had decided to spend a summer with Lord Byron, John Polidori, and Claire Clairmont. One night, they all took upon a challenge to create a horror story. Mary Shelley was the only one who was able to do so. Intrigued by Mary's tale, Percy pushed her to make the story more than it was. In the preface to the novel, Shelley even gives her husband full credit for the masterpiece.
As far as I can recollect, it was entirely written by him.
Therefore, Mary Shelley had no lack of influence. Both of her parents and her husband willed her to write.